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Given the set $S=\{0,1,2\}$, it has been asked to prove that it is not a group under the operation $\max(x,y)$. It can be done. Then they ask to identify $3$ subsets of $S$ which are 'groups' under the operation $\max(x,y)$.

How to choose the subsets?

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up vote 4 down vote accepted

Note that $\max(x,x) = x$ for all $x \in S$. So if $(S', \max)$ is a group, we must have $|S'| = 1$ (for otherwise there is no identity). That is $\{0\}$, $\{1\}$ and $\{2\}$ are the subsets looked for.

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can we have a single element with a BINARY operation and call it a group? – Fazlan Nov 21 '12 at 12:52
Yes ... the operation is then the(!) map $\circ \colon \{*\} \times \{*\} \to \{*\}$ – martini Nov 21 '12 at 13:08

Hint: A singleton is always a group.

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